Introduction of matrices

1. MATRICES

Tutorial Prepared
By
Narender Sharma
Owner , Manager and Blogger
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Promoting quality culture in every sphere of human life.

�
11�
12�
13
�
21�
22�
23
�
31�
32�
33



Introduction of Matrices
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2. Matrices 2
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S. No. Topic Page No.
1 Definition of Matrices 3-4
2 Notation of Matrices 5
3 Order of Matrices 6
4 Types of Matrices 7-11
5 Trace of Matrices 12
6 Equality of Two Matrices 13
7 MCQ‟s for Exercise 14-15
8 Answer Key 16
Index

3. A matrix is a rectangular array of numbers. In other words, a set of „m‟, „n‟
numbers arranged in the form of rectangular array of m rows and n column is
called m×n matrix read as „m‟ by „n‟ matrix.



A =
a
11a
12a
13………..
a
21a
22a
23………..
a
31a
32a
33………..
m×n



We can understand matrix by the fig. shown above, which shows a mess of
vertical and horizontal lines. The crossing points of vertical and horizontal
lines are the position of elements of matrix. In above fig there are 9 crossing
points which can be find out by multiplying no. of horizontal and vertical
lines i.e. 3×3=9.
Matrices 3
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Horizontal
lines
Vertical
lines
Definition of Matrix

4. The numbers a
11, a
12 ……..etc are called elements of the matrix A. „m‟ is the
numbers of rows and „n‟ is number of column.

e.g. A=
234
546
243
3×3

Matrices 4
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5. •Matrices are denoted by capital letters A, B, C or X , Y , Z etc.
•Its elements are denoted by small letters a, b, c ….. etc.
•The elements of the matrix are enclosed by any of the brackets i.e.
,, .
•The position of the elements of a Matrix is indicated by the subscripts
attached to the element. e.g. �
13 indicates that element „a‟ lies in first row
and third column i.e. first subscript denotes row and second subscript
denote column.
Matrices 5
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�
1 3
Element Indicate Row
Indicate Column
Notation of Matrices

6. •The number of rows and columns of a matrix determines the order of the
matrix.
•Hence , a matrix , having m rows and n columns is said to be of the order
m×n ( read as „m‟ by „n‟).
•In particular, a matrix having 3 rows and 4 columns is of the order 3×4
and it is called a 3×4 matrix e.g.

A=
235
467
is a matrix of order 2×3 since there are two rows and three
columns.

A=357 is a matrix of order 1×3 since there are one row and three
columns.

A=
5
6
8
is a matrix of order 3×1 since there are three rows and one
column.

Matrices 6
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Order of Matrices

7. Matrices 7
Rectangular
Matrix

Square
Matrix

Diagonal
Matrix

Scalar
Matrix

Unit
Matrix

Null
Matrix

Row
Matrix

Column
Matrix

Upper
Triangular
Matrix

Lower
Triangular
Matrix

Sub
Matrix

Types of Matrices
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8. Rectangular Matrix : A matrix in which the number of rows and
columns are not equal is called a rectangular matrix e.g. ,

A=
245
467
2×3 is rectangular matrix of order 2×3


Square Matrix : A matrix in which the number of rows is equal to the
number of columns is called a square matrix e.g.

�=
234
325
785


Note : The elements 2, 2 , 5 in the above matrix are called diagonal elements and the line along
which they lie is called the principal diagonal

Matrices 8
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Principal Diagonal

9. Diagonal Matrix : A square matrix in which all diagonal elements are
non- zero and all non-diagonal elements are zeros is called a diagonal
matrix e.g.

�=
200
020
005
is a diagonal matrix of 3×3

 Scalar Matrix : A diagonal matrix in which diagonal elements are equal
(but not equal to 1), is called a scalar matrix e.g.

�=
200
020
002
is a scalar matrix of 3×3

Identity (or Unit) Matrix : A square matrix whose each diagonal
element is unity and all other elements are zero is called and Identity (or
Unit) Matrix. An Identity matrix of order 3 is denoted by I
3 or simply by I.
Matrices 9
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10. e.g.
I
3=
100
010
001
is a unit matrix of order 3

Null (Zero) Matrix : A matrix of any order (rectangular or square) whose
each of its element is zero is called a null matrix (or a Zero matrix) and is
denoted by O. e.g.

O=
00
00
and O=
000
000
are null matrices of order 2×2 and 2×3
respectively.

Row Matrix : A matrix having only one row and any number of columns
is called a row matrix (or a row vector) e.g.

�=123 is a row matrix of order 1×3
Matrices 10
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11.  Column Matrix : A matrix having only one column and any number of
rows is called a column matrix (or a column vector) e.g.
A=
1
3
6
is a column matrix or order 3×1
 Upper Triangular and Lower Triangular Matrix: A square matrix is
called an upper triangular matrix if all the elements below the principal
diagonal are zero and it is said to be lower triangular matrix if all the
elements above the principal diagonal are zero e.g.
�=
234
015
007
, �=
500
870
431

Sub Matrix : A matrix obtained by deleting some rows or column or both
of a given matrix is called its sub matrix. e.g.
�=
234
515
397
, Now
23
39
is a sub matrix of given matrix A. The sub
matrix obtained by deleting 2
nd
row and 3
rd
column of matrix A.
Matrices 11
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LTM UTM

12. The sum of all the elements on the principal diagonal of a square matrix is
called the trace of the matrix. It is denoted by tr. A .

If A=
a
11a
12a
13
a
21a
22a
23
a
31a
32a
33
then

trace of A = tr.A = �
11+ �
22+ �
33

e.g. A=
234
515
397
, then

trace of A = tr. A =2+1+7=10
Matrices 12
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Trace of Matrix

13. Two matrices „A‟ and „B‟ are said to be equal if only if they are of the same
order and each element of „A‟ is equal to the corresponding element of „B‟.
Given
A =
a
11a
12a
13……..
a
21a
22a
23……..
a
31a
32a
33……..
m×n and B =
b
11b
12b
13………
b
21b
22b
23………
b
31b
32b
33………
x×y
The above two matrices will be equal if and only if
 No. of rows and column of A and B are equal i.e. m=x and n=y.
Each element of A is equal to the corresponding element of B i.e. a
11=b
11,
a
12=b
12, a
13=b
13 ……… and so on

Let �=
��
��
and �=
34
56
, according to the condition of equality of
matrices both matrices have same order i.e. 2×2 but the 2
nd
necessary
condition is a=3, b=4, c=5 , d= 6





Matrices 13
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Equality of Two Matrices

14. 1. If in a given matrix the rows are 2 and the column are 3 then maximum no. of elements in this
matrix are
A) 4 B) 6 C) 8 D) 10

2. Element a
32 is belongs to which row and column in a given matrix A
A) 2
nd
row, 3
rd
column
B) 3
rd
row , 2
nd
column
C) can’t decide
D) None of the above
3. What is the order of given matrix
245
757
903

57
81
24

A) 3×5 B) 5×3 C) 3×3 D) 5×5

4. A matrix which do not have equal no. of rows and columns called
A) Square Matrix B) Triangular Matrix C) Scalar Matrix D) Rectangular Matrix

5. A matrix in which the rows and columns are equal in no. is called
A) Square Matrix B) Triangular Matrix C) Scalar Matrix D) Rectangular Matrix

6. In a Diagonal matrix all the elements on the principal diagonal can’t be equal to
A) 0 B) 1 C) 2 D) 3






Matrices 14
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MCQ’s for Excercise

15. 7. In a scalar matrix all the diagonal elements are
A) Equal to one B) Equal but greater than one C) Not equal D) None of the above

8. In a Unit matrix
A) All element are zero B) All elements are one C) Diagonal elements zero rest are
one D) Diagonal elements are one rest are zero

9. In a given matrix all elements below principal diagonal are zero , the matrix is
A) Upper triangular matrix B) Lower triangular matrix C) Diagonal Matrix D) Scalar Matrix

10. In a given matrix all elements above principal diagonal are zero , the matrix is
A) Upper triangular matrix B) Lower triangular matrix C) Diagonal Matrix D) Scalar Matrix

11. Given A =
a
11a
12a
13
a
21a
22a
23
a
31a
32a
33
then tr. A will be
A) �
11+ �
12 + �
13 B) �
11+ �
22 + �
33 C) �
31+ �
22 + �
13 D) �
21+ �
22 + �
23

12. Which one is true about the necessary condition for equality of two matrices
A)Order of two matrices are same
B)Corresponding elements of the two matrices are same
C)Both A and B
D)None of the above


Matrices 15
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16. Matrices 16
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S.No. 1 2 3 4 5 6
Ans. B B A D A A
S.No. 7 8 9 10 11 12
Ans. B D A B B C
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Answer Key